Editing Support (mathematics) (section)

==Singular support==

In [[Fourier analysis]] in particular, it is interesting to study the '''{{em|{{visible anchor|singular support}}}}''' of a distribution. This has the intuitive interpretation as the set of points at which a distribution {{em|fails to be a smooth function}}.

For example, the [[Fourier transform]] of the [[Heaviside step function]] can, up to constant factors, be considered to be <math>1/x</math> (a function) {{em|except}} at <math>x = 0.</math> While <math>x = 0</math> is clearly a special point, it is more precise to say that the transform of the distribution has singular support <math>\{ 0 \}</math>: it cannot accurately be expressed as a function in relation to test functions with support including <math>0.</math> It {{em|can}} be expressed as an application of a [[Cauchy principal value]] {{em|improper}} integral.

For distributions in several variables, singular supports allow one to define {{em|[[wave front set]]s}} and understand [[Huygens' principle]] in terms of [[mathematical analysis]]. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).